Top Rigidity of Torus

8/19/2019 Top Rigidity of Torus

1/83


2/83

. . .

M

M

M

Sn

f : M → N

n ≥ 3 f

W h(Zn)


3/83

Zn

2

S P L(T

k × Dn)

4

5

P L

6


4/83

K 0, K 1

K 1

W h(Zn)

πi(PL/O)

,

J

πi(G/PL)


5/83

S P L(Tk × Dn), n + k ≥ 5

TOP/PL


6/83

τ (f ) ∈ W h(π1(N )) f : M → N

f

W h(π1(M ))

M

W h(Zn)

K

K 1

W h(Zn)

R

K 0, K 1

(i)

R

R

(ii)

R

R

R

K

K 0, K 1

P

A


7/83

P

i.e

0 → P 1 → P → P 2 → 0

A

P 1, P 2 ∈ P P ∈ P

P

i.e

P

P 0

P 0 → P

R

R

R

R

Rn, n ≥ 0 R

Rn, n ≥ 0

K

R

R

P

P 0

Λ0(P ) [P ], P ∈ P 0

K 0(P ) Λ0(P )

[P ] = [P 1] + [P 2] P

0 → P 1 → P → P 2 → 0.

P

P 0

[P ]

P ∈ P

P

P 0

Λ1(P ) (P, α), P ∈

P 0, α ∈ P K 1(P ) Λ1(P )

[P,αβ ] = [P, α] + [P, β ]

P

0 P 1ι

α1

P π

α

P 2

α2

0

0 P 1ι P

π P 2 0


8/83

α ∈

P

α1 ∈ P 1 α2 ∈ P 2

[P, α] = [P 1, α1] + [P 2, α2].

P P 0

[P, α]

P ∈

P

α ∈

P

K 1

K 1

n × n

I n + rE i,j r ∈ R E i,j = (δ ik,jl)1≤k≤n,1≤l≤n (1 ≤ i ≤ n, 1 ≤ j ≤ n).

GLn(R)

E n(R)

GLn(R) → GLn+1(R)

M →

M 0

0 1

,

E n(R) E n+1(R) GL(R) E (R)

GLn(R) E n(R)

E (R) = [GL(R), GL(R)]

(i)

K 1(R) = GL(R)/E (R) = GL(R)ab

(ii) π

W h(π)

K 1(Zπ) GL1(Zπ)

GL(Zπ) π

K 1(R) K 1(R) K 1(Proj R)

K i(R) = K i( R) i = 0, 1

K

P

M

F : P → M

F ∗ : K i(P ) → K i(M), i = 0, 1.


9/83

K

R

R

1.2

i∗ : K 1( R) → K 1(R

)

R

R

R

M

i.e

0 → P n → . . . → P 1 → M

P i R

R

R[t]

R R[t, t−1]

R[t, t−1]

R

M

R[t, t−1]

M

M 1 R[t]

0 → P n → . . . → P 1 → M 1

R[t]

R[t, t−1]

R[t]

R[t, t−1] =

lim t−nR[t]

t−nR[t]

R[t]

0 → R[t, t−1] ⊗R[t] P n → . . . → R[t, t−1] ⊗R[t] P 1 → R[t, t

−1] ⊗R[t] M 1 M

R[t, t−1]

M

Z[Zn]

n ≥ 0

Z

Z[Zn+1] Z[Zn][t, t−1]


10/83

K 1

R

K 1(R)

K 1( R), K 1(R ).

W h(Zn)

[ ]f g [ ] proj K ∗(R )

K ∗( R)

K 1(R ) → K 1( R)

R

R

R

R

M

R

α

M

0 → P r → . . . → P 1 → M

αi ∈ P i, 1 ≤ i ≤ r,

0 P r

αr

. . . P 1

α1

M

α

0

0 P r . . . P 1 M 0

P → M → 0 P P

M

P

M

P

α ⊕ α−1 ∈

M ⊕ M

α ⊕ α = α 00 α−1 = 1 α

0 1 1 0

−α−1 1 1 α

0 1 0 −1

1 0 .

α

α−1

β

β

P

1 β

0 1

1 0

−β 1

1 β

0 1

0 −1

1 0


11/83

α ⊕ α−1

α1 P ⊕ P

P ⊕ P π⊕0

α1

M

α

0

P ⊕ P π⊕0 M 0.

ker(π ⊕ 0)

α1 α1

ker(π ⊕ 0)

R

ker(π ⊕ 0) α1 ∈ ker(π ⊕ 0)

. . . dr+1 P r

dr

αr

. . . P 1

α1

M α

0

. . . dr+1 P rdr . . . P 1 M 0

αi ∈ P i, i ≥ 0 R

r

r

. . . dr+1 P rdr . . . P 1 M 0,

ker dr αr+1 ker dr

0 ker dr

αr+1

dr+1 P rdr

αr

. . . P 1

α1

M

α

0

0 ker drdr+1 P r

dr . . . P 1 M 0

Φ : K 1(R

) → K 1(

R)

[M, α] ∈ K 1(R )

0 P r

αr

. . . P 1

α1

M

α

0

0 P r . . . P 1 M 0.


12/83

Φ ([M, α]f g) =

i≥1(−1)i[P i, αi] proj ∈ K 1( R).

Φ

R

M, M

α :

M → M

0 → P r → . . . → P 0 → M → 0,

0 . . . P r+1

αr+1

P r

αr

. . . P 1

α1

M

α

0

0 . . . 0 P r . . . P 1 M 0.

M

α

P 0

d0 M 0

P 0

α0

M

α

P 0

d0 M 0.

B = ker P 0 ⊕ M

−d0⊕α−−−−→ d0 B → M

P 0 → B P 0

B → P 0 B → M

P 0d0

α0

M

α

0

P 0d0 M 0.


13/83

P

i

di

αi

. . . P

0

α0

M

α

P i+1 P i

di . . . M 0,

ker di

α

P i+1 ker di 0

P i = 0 i ≥ r + 1

ker dr+1

Φ([M, α]f g)

α

0 P r

αr

. . . P 1

α1

M

α

0

0 P r . . . P 1 M 0

0 P r

αr

. . . P 1

α1

M

α

0

0 P r . . . P 1

M 0.

M

∆

0 P r ⊕ P

r

dr⊕dr . . . P 1 ⊕ P 1

d1⊕d1 M ⊕ M 0,

∆

. . . → P 1 → M


14/83

f • : P

• → P • f

• : P

• → P

• M P •

. . . → P 2d2−→ P 1 → 0

P • P

•

M s

M

P 1

f 1

M

Id

0 P 1

f 1

M

Id

0

P 1 M 0 P 1

M 0,

f 1

f

1

f

f

Proj R

0 → Qn → . . . → Q1 → Q0 → 0,

χ(Q•) =

i(−1)i [Qi] proj

χ(C f ) = χ(P •) − χ(P • ) χ(C f ) = χ(P

•) − χ(P

• )

K 1 K 1(

R) ι∗−→ K 1(R )

R

P

0 → P → P → 0

Φ ◦ ι∗ ([P ] proj ) = Φ ([P ]f g) = [P ] proj

R

mod

0 → M n → . . . → M 1 → M 0 → 0,

χ(M •) =

i(−1)i [M i]f g

R

0 → P n → . . . → P 1 → M → 0,

ι∗ ◦ Φ ([M ]f g) = ι∗

i≥1

(−1)i+1[P i] proj

=i≥1

(−1)i+1[P i]f g = [M ]f g.


15/83

R

R = Z[Zn] Z → R

K 0(Z) ≈−→ K 0(R)

K 0(R)

≈−→ K 0(R[t, t

−1])

R

R

R

R[t]

R[t]

M

ϕi(M ) ⊂ M i M i =

j≥1 Rt

j M i− j ϕi(M ) = M i/Di(M ) ϕ(M )

R i ϕi(M )

M

ϕ(M ) = 0

M = 0

M

n

M n = 0 ϕn(M ) = 0 ϕn(M ) =

M n

ϕ

Q R[t] ⊗R Q

R

R[t]

ϕ

ϕ(R[t] ⊗R Q) Q R

R[t] ⊗R ϕ(P ) P R[t]

f : P → ϕ(P )

ϕ(P )

R

ϕ(P )

g : ϕ(P ) → P

R[t]

h : R[t] ⊗R ϕ(P ) → P

ϕ(h) : ϕ(R[t] ⊗R ϕ(P )) ≈−→ ϕ(P ) ϕ

ϕ(

h) = 0

h

h

h

P

h


16/83

ϕ

ϕ(ker h) ker(ϕ(R[t] ⊗R ϕ(P )) → ϕ(P )) = 0

ker h = 0

h

Q R[t, s] ⊗R Q

R

R[t, s]

K 0(R) → K 0(R[t])

Q R ⊗R[t] Q K 0(R[t]) → K 0(R)

R ⊗R[t] − R[t]

R ⊗R[t] M M/(t − 1)M

R[t]

M

M

(1 − t)M ∩ M = (1 − t)M

x = x0 + x1 + . . . ∈ M (1 − t)x =

x0 + (x1 − tx0) + . . . + (xn − txn−1) + . . . ∈ M

xn M

K 0(R) → K 0(R[t])

K

0(R[t])

→ K

0(R)

P

R[t]

R[t] ⊗R[t,s] N R[t, s] P =

R[t]n/Q

n ≥ 0

Q ⊂ R[t]n

M

R

Q

Q

f i = (f j,1(t), . . . , f j,n(t)) , 1 ≤ i ≤ m,

gi = (g j,1(t, s), . . . , g j,n(t, s)) , 1 ≤ i ≤ m,

atk

atksd−k

d

f i,j gi,j d

Q

R[t, s]

gi,j N = R[t, s]/Q

R[t] ⊗[t,s] N P


17/83

R

R[t, s]

0 → P m → . . . → P 1 → N → 0.

0 → R[t] ⊗R[t,s] P m → . . . → R[t] ⊗R[t,s] P 1 → R[t] ⊗R[t,s] N P → 0

P i

R[t, s] ⊗R Q Q R R[t] ⊗R[t,s]

(R[t, s] ⊗R Q) R[t] ⊗R Q R[t] ⊗R[t,s] P i

R[t] ⊗ Q

0 → P m → . . . P 0 → 0

R

i

(−1)i [P i ] proj = 0 ∈ K 0(R).

K 0(R) → K 0(R[t, t−1])

R

R[t, t−1] → R

t

1

K 0(R) → K 0(R[t, t

−1])

K 0(R[t]) R → R[t] →

R[t, t−1]

K 0(R[t]) → K 0(R[t, t−1])

P

R[t, t−1]

P = R[t, t−1]n/Q

Q ⊂ R[t, t−1]n P R

Q

d tdQ ⊂ R[t]n

P tdR[t, t−1]n/

tdQ

R[t, t−1] ⊗R[t] M

R[t]

M

0 → P m → . . . → P 1 → M → 0


18/83

R[t]

R[t, t−1]

R[t]

0 → R[t, t−1] ⊗R[t] P m → . . . → R[t, t−1] ⊗R[t] P 1 → R[t, t−1] ⊗R[t] M P → 0.

R[t, t−1]

K 0(Z) → K 0(Z[Z

n])

n ≥ 0

W h(Zn)

[ ]W h

W h(Zn)

n ≥ 0

n

W h(e)

n ≥ 0

[P, α]W h ∈ W h(Zn+1)

R[t, t−1]

Q

P ⊕ Q R[t, t−1]N N ≥ 0

W h

[P, α]W h = [P ⊕ Q, α ⊕ IdQ]W h = [R[t, t−1]N , β ]W h

β

R[t, t−1]

R[t, t−1]N

[R[t, t−1]N , β ] = 0 ∈ W h(Zn+1)

β ∈

R[t, t−1]N

N ≥ 0

R[t, t−1]N

GLN (R[t, t

−1])

GLN (R[t, t

−1])

B ∈ GL(R[t, t−1]) GL(R)

E (R[t, t−1])

tm 0

0 1

(1 + A(t − 1)),

m ∈ Z

A ∈ M (R)

A(1 − A)


19/83

m ≥ 0

tmB

R[t]

tmB = B0 + tB1 + . . . + tdBd,

Bi R d ≥ 0

GL(R)

E (R[t])

d ≤ 1 d > 1

M ≈ N

M, N ∈ M (R[t, t−1])

GL(R)

E (R[t])

tmB ≈

tmB 0

0 1

≈

tmB td−1Bd

0 1

≈ tmB − tdBd td−1Bd

−t 1 ,

≤ d−1

tmB ≈ B0 + tB1 = (B0 + B1) + (t − 1)B1

tmB

R[t, t−1] B0+N 1

B0 + B1 ∈ GL(R) 1 + A(t − 1) = (1 − A) + tA

C −r, . . . , C s ∈ M (R) s, r > 0

1 = ((1 − A) + tA)(t−rC −r + . . . + tsC s) = (C 0 + tC 1 + . . . + t

sC s)((1 − A) + tA).

(1 − A)C i + AC i−1 = 0

i = 0

AC s = 0

AiC s−i+1 = 0 1 ≤ i ≤ s + 1 A

s+1C 0 =

0.

(1 − A)C −r = 0 (1 −

A)iC −r+i−1 = 0 1 ≤ i ≤ r (1 − A)rC −1 = 0.

(1 − A)C 0 + AC −1 = 1 A

s(1 − A)r−1

As(1 − A)r−1 = (1 − A)r−1As+1C 0 + As(1 − A)rC −1 = 0,

[R[t, t−1]N , B]W h = [R[t, t−1]k, (1−A)+tA]W h +[R[t, t−1]k, S ]W h

+[R[t, t−1]k, U ]W h ∈ W h(Zn+1),

A(1 − A)

S ∈ GLm(R) U ∈ E (R[t])

[R[t, t−1]m, S ]W h = 0 W h(Z

n) = 0

S ∈ GLk(Z[Zn])

E (R[t])


20/83

P

R[t, t−1]

α

P

[P, α] proj = 0 ∈ K 1(R[t, t

−1])

R[t, t−1

]

αs = 0

M i = (α−1)

s−i

M i α

α

M i+1/M i

R[t, t−1]

0 M i

α|

M i+1

α|

M i+1/M i

Id

0

0 M i M i+1 M i+1/M i 0.

K 1 [M i+1, α|]f g = −[M i, α|]f g

[R[t, t−1]k, 1+(t−1)A]W h = 0

A ∈ M (R) A(1 − A) As(1 − A)s = 0

R[t, t−1]k = M 0 ⊕ M 1 M 0 = ker As, M 1 = K er(1 − A)

s

A

A0 A1 M 0 M 1

R[t, t−1]k, 1 + (t − 1)A

W h = [M 0, IdM 0 + (t − 1)A0]W h

+ [M 1,tIdM 1 + (IdM 1 − tA1)]W h

= [M 0, IdM 0 + (t − 1)A0]W h + [M 1, ]W h+

IdM 1 + t−1A−11 (IdM 1 − tA1)

W h

= [M 1,tIdM 1]W h + [M 0, IdM 0 + (t − 1)A0]W h

+ [M 1, A1]W h +

IdM 1 + t−1A−11 (IdM 1 − tA1)

W h

M 1 [M 1,tIdM 1]W h

M 1

[M 1,tIdM 1 ]W h = 0

s ∈ Z

[M 1] proj = [R[t, t

−1]s] proj

K 0( R[t, t

−1])

M 1

Q

Q = R[t, t−1]s +

i

(P (i) − P (i)1 − P

(i)2 ) ∈ Λ0( R[t, t

−1]),


21/83

P (i), P

(i)1 , P

(i)2

0 → P (i)1 → P

(i) → P (i)2 → 0.

0 P (i)1

tId

P (i)

tId

P (i)2

tId

0

0 P (i)1

P (i) P (i)2

0.

(Q,tId) = R[t, t−1]s,tId+i(P (i),tId) − (P (i)1 ,tId) − (P (i)2 ,tId) ∈ Λ1( R[t, t−1]),

[M 1,tId]W h = [R[t, t

−1]s,tId]W h = 0.


22/83

f : M → M N

M

f

N

g : N → N N = f −1(N ) f

g

g

N

M

2.2

f : M → Tn

n ≥ 6 N Tn

f

f

N

f | : N → N

N = f −1(N )


23/83

R

Z[Zn]

M

n ≥ 6 f : M → Tn

N

M = Tn

f

N

g : N →

N

N = f −1(N )

N

g

f

N

g : N → N

π1

N

N

g : N → N

g∗ : π1(N ) → π1(N

)

Z → N g∗π1(N )

g

g̃ : N → Z

g∗ : H n−1(N ) → H n−1(N )

H n−1(N ) g̃∗−→ H n−1(Z ) → H n−1(N

)

g

Z

H n−1(Z ) = 0) g∗π1(N ) π1(N

)

d

H n−1(Z ) → H n−1(N )

d

d = 1

g∗π1(N ) = π1(N

)

γ

ker g∗ f : M → M

π1(N )

g∗

π1(M )

f ∗≈

π1(N

)

π1(M )

γ

M

n ≥ 6

γ

(D2, S1) → (M, N )


24/83

D2

D2 × Dn−2 f

f

N

N 1

γ

N 1 = N − (S1 × D

n−2) ∪ D2 × Sn−3

f

g1 : N 1 → N

ker g1 ker g∗/ < [γ ] >

D2 × Dn−2

N

g1

N

N → N

π1

πi, i ≥ 2

H i, i ≥ 1

h : X → Y

m

H i(X ) f ∗

D

H i(Y )

D

H m−i(X ) H m−i(Y )f ∗


25/83

D

f = D

−1 ◦ f ∗ ◦ D

y ∈ H i(Y )

f ∗ ◦ f (y) = f ∗(f ∗(Dy) ∩ [X ]) = Dy ∩ f ∗[X ] = Dy ∩ [Y ] = y.

f ∗ ◦ f = id f ∗

f

π1

f

g

N 1 N

N → N 1

p : Y M → M

Zn−1 = π1(N ) → π1(M

) = Zn p : Y M → M

f

Y M

p

f Y M

p

M

f M .

p N

N

N

Y M AN BN

Z

t

tAN ⊂ AN π1(N ) → π1(N )

N

N

f (N ) ⊂ N N Y N AN BN

t

p

π1(N ) → π1(N

)

f (AN ) ⊂ AN f (BN ) ⊂ BN

tAN ⊂ AN


26/83

h : X → Y K i(h) = ker H i(X ) → H i(Y )

K i(h) = ker H i(Y ) → H i(X )

i ≥ 0

K i(X ) K

i(X )

i ≥ 1

K i(N ) K i−1(AN , N ) ⊕ K i−1(BN , N )

R

→ K i+1(Y M ) → K i(N ) → K i(AN ) ⊕ K i(BN ) → K i(Y M ) →

→ K i(Y M ) → K i(Y M , AN ) → K i−1(AN ) → K i−1(Y M ) →

→ K i(Y M ) → K i(Y M , BN ) → K i−1(BN ) → K i(Y M ) →

K i(Y M , AN ) K i(BN , N ) K i(Y M , BN ) K i(AN , N )

f

K i(Y M ) = 0 i ≥ 0


27/83

(Dk, Sk−1) → (N, AN − tAN )

AN − tAN

Dk

Dk × Dn−k f f

N

N

N 1

N 1 = N −(Sk−1×Dn−k)∪Dk ×Sn−k−1

N 1 M N

W

N

N 1

f

g1 : N 1 → N

K i(AN , N )

K i(BN , N )

g

k

K k(N ), K k+1(AN , N ) K k+1(BN , N )

R

K k(N ) H k(N ) → H k(N )

K i(N ) H i+1(C g) i ≥ 0 C g


28/83

g

g

k

πi(C g) = 0 = H i(C g) i ≤ k

H k+1(C g) R

t

i : (Y M , BN ) → (Y M , tBN )

R

(t−1)∗ : H i(AN , N ) → H i(AN , N )

H i(Y M , BN )

≈

i∗ H i(Y M , tBN , )(t−1)∗ H i(Y M , BN )

≈

H i(AN , N )

(t−1)∗ H i(AN , N ),

t∗ : H i(BN , N ) → H i(BN , N )

K i(AN , N ) (K i(BN , N )

g

k

(t−1)∗ t∗ K k+1(AN , N )

K k+1(BN , N )

(t−1)∗ t∗

x

(t−1)∗ c

x

c

l

c

AN − tlAN x H i(Y M , BN ) → H i(Y M , tlBN )

(t−1)l∗x = 0

g

k

l ≥ 1

(t−1)∗ K k+1(AN , N )

πk+1(AN − tAN , N ) H −→ H k+1(AN − tAN , N )

j∗−→ H k+1(AN , N ),

H

j∗

(t−1)l−1∗

f

N ⊂ M

g = f |N : N → N

k

k < n/2


29/83

N ⊂ M

g

π1

k

k = 0, 1

g

k

k < n/2

k + 1 > n/2

K k+1(AN , N )

R

R

(t−1)∗

l

(t−1)∗K k+1(AN , N ) x1, . . . , xs

πk+1(AN − tAN , N )

x1, . . . , xs k + 1 < n/2

(Dk+1, Sk) → (AN − tAN , N )

f

f

N 1 ⊂ N AN 1 BN 1 N 1

K k+1(AN , N )

≈

K k+1(AN 1, N 1)

≈

K k+1(Y M , BN ) K k+1(Y M , BN 1) K k(BN 1, BN ).

K k(BN 1 , BN ) K k(W, N ) W

N

N 1 W N k + 1

K k(W, N ) = 0 K k+1(AN , N ) → K k+1(AN 1, N 1)

K k+1(Y M , BN ) → K k+1(Y M , BN 1)

(t−1)l−1∗ (K k+1(AN , N ))

K k+1(AN , N )

(t−1)∗

K k+1(AN 1 , N 1)

(t−1)∗

0

K k+1(AN , N ) K k+1(AN 1 , N 1) 0,

(t−1)∗K k+1(AN 1 , N 1) = 0

K k+1(AN , N )

K k+1(BN , N ) W N 1

n − k − 1 n − k − 1 > n/2 k + 1 < n/2

K k+1(Y M , AN 1) → K k+1(Y M , AN ) K k+1(BN , N )

K k+1(BN 1, N 1) K k+1(BN , N )


30/83

(i)

n = 2k

f N

g : N → N K i(AN , N ) = 0, K i(BN , N ) = 0 i < k

K k(AN , N ) = 0 K k(BN , N )

(ii)

n = 2k + 1

f

N

g : N → N

K i(AN , N ) = 0, K i(BN , N ) = 0 i ≤ k K k+1(AN , N )

K k+1(BN , N )

(i)

K i(AN , N ) = 0, K i(BN , N ) = 0 i <

k

x1, . . . , xs

π1(N ) → π1(AN − tAN )

4

K k(AN , N )

K i(AN , N, R) ⊕ K i(BN , N, R) = K

i−1(N, R)

R

R

K i−1

(N, R) = 0

i > k

K i(BN , N, R) = 0 i > k K k(BN , N )

R

(ii)

K i(AN , N ) = 0, K i(BN , N ) = 0 i ≤ k

K i−1(N, R) = 0

i > k + 1

K i(AN , N, R) ⊕

K i(BN , N, R) = 0 i > k + 1 K i(AN , N, R) = 0 K

i(BN , N, R) = 0

i > k + 1

K k+1(AN , N, R) K k+1(BN , N, R)

R

M → M

K 0(R) Z

R

R

R

(P, ν )

P

R

ν

P

R = K 0( R).


31/83

K 0(R) Z

R

(K k+1(BN , N ), t∗) (K k(BN , N ), t∗) R n = 2k +1

n = 2k

(P, ν )

Nil R

0 = E 0 ⊂ E 1 ⊂ . . . ⊂ Rr = P E i+1/E i

ν (E i+1) ⊂ E i i

P

R

P ⊗R R[x]/(x

r)

ν (

i pi ⊗ xi) =

i pi ⊗ xi+1

Nil R

(P, ν )

1

Nil R

m − 1, m ≥ 2

(P, ν )

0 = E 0 ⊂ E 1 ⊂ . . . ⊂ E m = P

R

0 → (E m−1, ν E m−1) → (P, ν ) → (P/E m−1, 0) → 0.

[P, ν ] = [E m−1, ν E m−1] + [P/E m−1, 0] = [E m−1, ν E m−1] P/E m−1

(E m−1, ν E m−1) m − 1

[E m−1, ν E m−1] = 0

(P, ν )

R

0 = E 0 ⊂ E 1 ⊂

. . . ⊂ E m = P ν (E i+1) ⊂ E i

R

0 → (P , ν ) u−→ (P , ν )

v−→ (P, ν ) → 0

(P , ν )

0 = F 0 ⊂ F 1 ⊂

. . . ⊂ F m = P

v(F i) = E i


32/83

m

P

m = 1

F

v : F → P

0 → (ker v, 0) → (F, 0) v−→ (P, 0) → 0

m − 1, m ≥ 2

(E m−1, ν E m−1)

vm−1 : (F m−1, f m−1) → (E m−1, ν E m−1)

E m/E m−1

q : Q → E m/E m−1 Q R

q

q̄ : Q → E m = P F = F m−1 ⊕ Q v = vm−1 ⊕ q : F → P

f m−1 F Q

f̄

Q

v

f̄ F m−1

vm−1

K ν

E m−1.

f = f m−1 ⊕ f̄ : F → F L = ker v l = f |L 0 → (L, l) → (F, v)

v−→

(P, ν ) → 0

R

[P, ν ] ∈ Nil R ν m = 0 K i = ν m−i

0 = K 0 ⊂ K 1 ⊂ . . . ⊂ K m = P K i

0 → (P 1, ν 1)

u

−→ (P

, ν

)

v

−→ (P, ν ) → 0

R

(P , ν )

0 = E 0 ⊂ E 1 ⊂ . . . ⊂ E m = P

v(E i) ⊂ K i (P

, ν )

[P, ν ] = −[P 1, ν 1]

Li = u−1(E i) 0 = L0 ⊂ E 1 ⊂ . . . ⊂ Lm = P 1

0 → Li+1/Li → E i+1/E i → K i+1/K i → 0. R

Li+1/Li Li

M

R

d(M )

M

E i+1/E i 0 →

Li+1/Li → E i+1/E i → K i+1/K i → 0 d(Li+1/Li) = (1, d(K i+1/K i) − 1)

d = max0≤i≤m−1d(K i+1/K i) d

(P d, ν d) ∈ R [P d, ν d] = (−1)d[P, ν ]

0 = F 0 ⊂ F 1 ⊂ . . . ⊂ F m = P d S i+1/S i

R

[P, ν ] = (−1)d[P d, ν d] =

i

[S i+1/S i, 0] = 0,

S i+1/S i


33/83

(K k(BN , N ), t∗) (K k+1(BN , N ), t∗)

Nil R = K 0( R)

0

K 0(R) NilR = K 0( R)

(P, ν ) ∈

R

0 ∈ Nil R

(T 1, t1), (T 2, t2)

(P, ν ) ⊕ (T 1, t1) (T 2, t2).

f

(i)

n = 2k ≥ 6

(N, g)

R

0 → (P, ν ) → (P 1, ν 1) → (F, f ) → 0

(P, v) (K k(BN , N ), t∗) (F, f )

(N 1, g1) (K k(BN 1 , N 1), t∗) (P 1, ν 1)

(ii)

n = 2k + 1 ≥ 7

(N, g)

R

0 → (P, ν ) → (P 1, ν 1) v−→ (F, f ) → 0

(P, v) (K k+1(BN , N ), t∗) (F, f )

(N 1, g1) (K k+1(BN 1, N 1), t∗) (P 1, ν 1)


34/83

n = 2k

(F, f )

(F, f ) (R, 0) a ∈ P 1

R

x = ν 1(a) ∈ ker v

(P, ν ) (K k(BN , N ), t∗) u = ∂x

. . . → K k(t−1BN , t

−1N ) K k(BN , BN − t−1BN ) ∂ −→

∂ −→ K k−1(BN − t−1BN , N ) → K k−1(BN , N ) 0 → . . .

u : (Dk−1, Sk−2) → (BN − t

−1BN , N )

u

u (N 1, g1) W

N

N 1

(BN , W, N )

0 K k(W, N ) → K k(BN , N ) → K k(BN , W ) K k(BN 1, N 1) →

→ K k−1(W, N ) → K k−1(BN , N ) 0,

K k(BN 1, N 1) K k(BN , N ) ⊕ R

u

t−1∗ x t∗(t−1∗ x) = x

t∗

0 → (K k(BN , N ), t∗) → (K k(BN 1 , N 1), t∗) → (K k−1(W, N ), t∗) → 0,

0 → (P, ν ) → (P 1, ν 1) → (R, 0) → 0

K k(AN , N ) (AN 1 , W, N 1)

n = 2k ≥ 6 (N, g)

R

0 → (F, f ) → (P, ν ) → (P 1, ν 1) → 0

(P, v) (K k(BN , N ), t∗) (F, f )

(N 1, g1) (K k(BN 1 , N 1), t∗) (P 1, ν 1)


35/83

(F, f ) (R, 0)

a

K k(BN , N ) P F l tl∗K k(BN , N ) =

0

t∗a = 0 a ∈ ker t∗ ⊂ tl−1

∗

a

(N 1, g1)

W

N

N 1

(BN , W, N )

0 → K k(W, N ) → K k(BN , N ) → K k(BN , W ) K k(BN 1, N 1) → 0,

t∗

0 → (R, 0) → (P, ν ) → (P 1, ν 1) → 0.

(AN 1, W, N 1)

K k(AN , N )

n = 2k + 1 ≥ 7

(N, g)

R

0 → (F, f ) → (P, ν ) → (P 1, ν 1) → 0

(P, v) (K k+1(BN , N ), t∗) (F, f )

(N 1, g1) (K k+1(BN 1, N 1), t∗) (P 1, ν 1)

4

(K k(BN , N ), t∗) (K k+1(BN , N ), t∗)

(N 1, g1) K k(BN , N ) = 0

K k+1(BN , N ) = 0

g1


36/83

R

K 1(R[t, t−1]) K 1(R) ⊕ K 0(R) ⊕Nil(R) ⊕Nil(R).

π

W h(π × Z) W h(π) ⊕ K 0(Zπ) ⊕N il(Zπ) ⊕Nil(Zπ).

f : M → M

m

m ≥ 6 π × Z

N

M

f

N

φ(τ )

τ

f

φ : W h(π × Z) → K 0(π) ⊕N il(R)

W h(π × Z)

φ(τ )

N il(Zπ)

N

Nil(Zπ)

Wh(π × Z)


37/83


38/83

[X ] ∈

H m(X ; Z)

∩[X ] : H n(X ; Z) → H m−n(X ; Z)

n

X

f : M → X M

f

π1

ker(f ∗ : π1(M ) → π1(X )) f 1 : M 1 → X

π1

πi, i ≥ 2

H i, i ≥ 2 f 1

π1 H i, i ≥ 2

H i(N )

f ∗−→ H i(X )

f : M → X

M

m

m = 2n

2n + 1

n

f n : M n → X

ker f ∗ : H n(M ) → H n(X ) L

Lm (Z[π1(X )])

n + 1

f n+1 : M n+1 → X

Ln(e) =

Z

n ≡ 0 (mod 4),

Z2 n ≡ 2 (mod 4)

0

n ≡ 1, 3 (mod 4)


39/83

(f, b) : N → M

M

S (f, b) =

18(σ(N ) − σ(M ))

n ≡ 0 (mod 4),

n ≡ 2 (mod 4)

0

n ≡ 1, 3 (mod 4).

M

n ≥ 5

W h(π1(M )) = 0

ξ ∈ Ln+1(π1(M ))

M

n ≥ 5

ξ ∈ Ln+1 (Z [π1(M )])

(F, b) : N → M × [0, 1]

F = (F ;0 F, ∂ 1F ) : (N ; ∂ 0N, ∂ 1N ) → (M × [0, 1]; M × 0 ∪ ∂M × [0, 1], M × 1)

∂ 0F

∂ 1F

ξ = S (F, b)

(ψ, b) : (N,∂N ) → (M × [0, 1]), ∂ (M × [0, 1])

ξ

L

M

ψ

L × [0, 1]

(ψ|, b|) : ψ−1(L × [0, 1]) → L × [0, 1].

Ln(G)

α(L) : Ln+1(G × Z) → Ln(G)

α(L)

Ln+1(G) Ln+1(i∗)−−−−−→ Ln+1(G × Z)

α(L)−−→ Ln(G)

S1


40/83

Ln(Zk) 0≤l≤k

k

lLn−l(e).

Ln+1(i∗)

ξ ∈ Ln+1(G)

(ϕ, b) : N → L × [0, 1] × [0, 1]

ξ

∂N =

∂ −N ∪ ∂ +N ϕ| : ∂ −N → (L × [0, 1] × {0}) ∪ (L × {0} × [0, 1]) ∪ (L × {1} × [0, 1])

L × {0} × [0, 1]

L × {1} × [0, 1] ∂N

(g, b) : N → L × S1 × [0, 1].

ι(L) : Ln+1(G) → Ln+1(G × S

1)

ι(L)

ι(L) = Ln+1(i∗)

(ϕ, b) : M → N

X

(ϕ × IdX , b × Id) : M × X → N × X.

S (ϕ × IdN )

X

Ln(π)

Li(π) ⊗ L j(π) → Li+ j(π × π)

S (ϕ × IdX ) = S (ϕ) ⊗ σ

∗(X )

σ∗ : Ω∗Bπ → L∗(π)


41/83

(i)

L∗(e)

4

Ln(e) = Z n ≡ 0 (mod 4),

Z2 n ≡ 1 (mod 4)

0

n ≡ 1, 2 (mod 4).

(ii) N

S (ϕ×IdN ) =

S (ϕ).σ(N )

dim N ≡ 0 (mod 4),

0

dim N ≡ 1 (mod 4)

0

dim N ≡ 2, 3 (mod 4).

Ln(π)

Ln( prπ)

⊗ σ∗(N ) Ln+m(π × π)

Ln+m( prπ×π)

Ln(e) Ln+m(e),

N

m

M

· · · → S CAT

M ×Dk+1, M ×Sk

→

M ×Dk, M ×Sk

, (G/CAT, ∗) S

−→ Lm+k (Z (π1(M ))) → · · ·

· · · → Lm+1 (Z [π1(M )]) → S CAT (M ) → [M,G/CAT ] S −→ Lm (Z [π1(M )]) .

M

M

m

S CAT (M,∂M )

(N, f )

N

m

f : N → M

∂f : ∂N → ∂ M


42/83

(N 1, f 1) (N 2, f 2)

h : N 1 → N 2

N 1

h

f 1 M

N 2.f 2

ν N

b η

N

f M,

f : N → M

η : M →

−

b : ν N → η (f, b) :

M → N f : N → M

[M,G/CAT ]

M

BCAT

M 0

BG


43/83

S

f : N → M m

m2

f : N → M S M (f, b) S (f, b)

Lm (Z [π1(M )])

S (M ) → [M,G/CAT ]

f : N → M

ν N

(f −1

)∗

ν N

N

f M.

Lm+1 (Z [π1(M )]) S (M )

f : N → M m

ξ ∈ Lm+1 (Z [π1(M )])

(F, b) : (W,∂W 0, ∂W 1) → (N ×[0, 1] , N ×{0} , N ×{1})

ξ

F |∂W 0 : ∂W 0 → N

ξ. (f : N → M ) =

f ◦ F −1|∂W 1 : ∂W 1 → N

[M,G/CAT ]

Lm (Z [π1(M )])

S

S

M

M × Dk

(k ≥ 1)


44/83

BG,BPL,BO,

G/PL PL/O

πi(PL/O)

P L

M

M →

→ →

7

≤ 5

M →

ωi ∈ H

i+1(M ; πi(PL/O)) M

≤ 5 ωi = 0 i ≥ 5 πi( ) = 0 i ≤ 4 ωi = 0 i ≤ 4

,

J

J

J :

πi(BO) → πi(BG) Si

i πi(BO) πi(BG) πi( ) J −→ πi( )

1 Z2 Z2 Z2 ≈−→ Z2

2 Z2 Z2 Z2≈

−→ Z2

3 0 Z2 0 0−→ Z2

4 Z Z24 Z pr−→ Z24

5 0 0 0 0−→ 0

6 0 0 0 0−→ 0


45/83

πi(G/PL)

πi(G/PL) → Li(e)

i ≥ 6 S P L(S

i) = 0.

πi(G/PL) → Li(e)

i ≥ 6

πi( ) Li(e)

Li(e), i ≤ 5

PL/O

7

πn(PL/O) → πn(G/O) → πn(G/PL) → πn−1(PL/O)

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